Definition 3.1.4 (The K_0 group of a Unital C*-algebra)

Let be a Unital -algebra, and let be the abelian semigroup from Definition 2.3.3. (The semigroup of projections) Define to be the Grothendieck group of . i.e. Define by Where is the Grothendieck map.
So take any projection p, and send it to its K_0 class via the Grothendieck map on its equivalence class inside of the semigroup of projections .